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Lecture 3, in class practice
Chebyshev’s Rule
k
1.5
2.0
2.3
3.0
1-1/k^2
0.56
0.75
0.81
0.89
1) In a random sample of the lifetime of an automobile battery, sample mean
is 54, sample standard deviation is 5.3. Use Chebyshev’s rule with k=2
and k=3 to describe the distribution of the battery lifetimes.
2) Suppose that in a random sample of songs produced by hard rock bands,
the sample mean is 3.35 (min) and sample standard deviation is 0.5.
a) Find the approximate proportion of observations between 2.35 and 4.35
minutes.
b)Find the approximate proportion of observations less than 1.85 or greater
than 4.85 minutes.
c) Approximately what proportion of songs lasts more than 5 minutes?
The two problems above are using the Chebyshev’s rule.
The next two problems are using empirical rule.
Chebyshev’s rule is applicable for all samples and all populations. So
the tail proportion is only an upper bound, and the bound is very
conservative. For a specific sample or population, the actual tail
proportion could be much smaller than the Chebyshev’s bound.
Empirical rule is for bell shaped and symmetric distributions. So if the
sample or the population is approximately normal, then we can apply
the rule. Note that the empirical rule gives much more accurate smaller
tail proportions for normally distributed data.
3) Over a long weekend in August 2012, there were 3556 speeding tickets
issued in Alberta. In a random sample of the tickets, suppose the shape of
the distribution is approximately normal, with sample mean 130$ and
sample standard deviation 25$.
a) Approximately what proportion of observation is between 80 and 180?
b) Approximately what proportion of observation is greater than 205 or less
than 55?
c) Approximately what proportion of observation is greater than 205?
d) Approximately what proportion of observation is between 105 and 180?
4) A company just developed a pain-relief medicine. The company claims
that the pain-relief time is hours is approximately normal with mean 8 and
standard deviation 0.2. A random patient is selected and his time is 7
hours. Is there any evidence to refute the claim?